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Chapter 6: Problem 41

Determine the following standard normal (z) curve areas: a. The area under the \(z\) curve to the left of 1.75 b. The area under the \(z\) curve to the left of -0.68 c. The area under the \(z\) curve to the right of 1.20 d. The area under the \(z\) curve to the right of -2.82 e. The area under the \(z\) curve between -2.22 and 0.53 f. The area under the \(z\) curve between -1 and 1 g. The area under the \(z\) curve between -4 and 4

Short Answer

Expert verified
The respective areas under the standard normal curve are a) 0.9599, b) 0.2483, c) 0.1151, d) 0.9974, e) 0.6864, f) 0.6826, and g) close to 1.

Step by step solution

01

Understanding Normal Distribution

Normal Distribution is a statistical concept that denotes the data distribution in the shape of a bell curve or 'normal curve'. This curve is symmetrical and its mean, median, and mode are equal. The standard normal distribution is a special case with mean equals 0 and standard deviation equals 1. The values of Z represent the number of standard deviations from the mean.
02

Using Z-score Table to Find Areas

A standard Normal Table, also called Z-table, is often used to find areas under the curve (probabilities) for Standard Normal Distbriution. Each value in the table presents cumulative probabilities up to the given Z-score. The positive values usually give the area to the left of the Z-score, and the area to the right can be calculated as 1 subtracted by the area to the left. The area between two Z-scores can be found by subtracting the area of the smaller Z-score from the bigger one.
03

Detailed Calculations

Use the Standard Normal Distribution Table (Z-table) to look up the areas:\na. The area under the Z curve to the left of 1.75 = 0.9599\nb. The area under the Z curve to the left of -0.68 = 0.2483\nc. The area under the Z curve to the right of 1.20 = 1 - 0.8849 = 0.1151\nd. The area under the Z curve to the right of -2.82 = 1 - 0.0026 = 0.9974\ne. The area under the Z curve between -2.22 and 0.53 = 0.6996 - 0.0132 = 0.6864\nf. The area under the Z curve between -1 and 1 = 0.8413 - 0.1587 = 0.6826\ng. The area under the Z curve between -4 and 4 = fails to 0 due to not finding the probability of Z > 4, so we can assume it is close to 1.

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Most popular questions from this chapter

Seventy percent of the bicycles sold by a certain store are mountain bikes. Among 100 randomly selected bike purchases, what is the approximate probability that a. At most 75 are mountain bikes? (Hint: See Example 6.33 ) b. Between 60 and 75 (inclusive) are mountain bikes? c. More than 80 are mountain bikes? d. At most 30 are not mountain bikes?

The Wall Street Journal (February \(15,\) 1972) reported that General Electric was sued in Texas for sex discrimination over a minimum height requirement of 5 feet, 7 inches. The suit claimed that this restriction eliminated more than \(94 \%\) of adult females from consideration. Let \(x\) represent the height of a randomly selected adult woman. Suppose that \(x\) is approximately normally distributed with mean 66 inches (5 ft. 6 in.) and standard deviation 2 inches. a. Is the claim that \(94 \%\) of all women are shorter than 5 feet, 7 inches correct? b. What proportion of adult women would be excluded from employment as a result of the height restriction?

The light bulbs used to provide exterior lighting for a large office building have an average lifetime of 700 hours. If lifetime is approximately normally distributed with a standard deviation of 50 hours, how often should all the bulbs be replaced so that no more than \(20 \%\) of the bulbs will have already burned out?

A machine that produces ball bearings has initially been set so that the mean diameter of the bearings it produces is 0.500 inches. A bearing is acceptable if its diameter is within 0.004 inches of this target value. Suppose, however, that the setting has changed during the course of production, so that the distribution of the diameters produced is now approximately normal with mean 0.499 inch and standard deviation 0.002 inch. What percentage of the bearings produced will not be acceptable?

Determine each of the following areas under the standard normal (z) curve: a. To the left of -1.28 b. To the right of 1.28 c. Between -1 and 2 d. To the right of 0 e. To the right of -5 f. Between -1.6 and 2.5 g. To the left of 0.23
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